A Fugue in Two Colors

Transcription

1 A Fugue in Two Colors Simon Hands (Swansea U.) Why two colors? Equation of state for µ Quark number susceptibility Topology Quarkonia Collaborators: Seyong Kim, Jon-Ivar Skullerud, Phil Kenny, Peter Sitch, Pietro Giudice, Alessandro Amato,Tim Hollowood, Joyce Myers STRONGnet, ECT* Trento 4 th October 2

2 The QCD Phase Diagram T(MeV) 2 T c RHIC/ALICE crossover (µ!," )! critical endpoint FAIR? quark!gluon plasma hadronic fluid nuclear matter crystalline quark matter? compact stars µ 5 onset 5 color superconductor µ (MeV)

3 František Kupka Fugue in Two Colors (92)

4 . p.9/3 The Sign Problem for µ In Euclidean metric the QCD Lagrangian reads L QCD = ψ(m + m)ψ + 4 F µνf µν Straightforward to show with M(µ) = D/ [A] + µγ γ 5 M(µ)γ 5 M ( µ) detm(µ) = (detm( µ)) ie. Path integral measure is not positive definite for µ Fundamental reason is explicit breaking of time reversal symmetry Monte Carlo importance sampling, the mainstay of lattice QCD, is ineffective

5 . p./3 What goes wrong with the usual positive HMC measure? detm M { M describes quarks q 3 M describes conjugate quarks q c 3 In general qq c gauge singlet bound states with B > In QCD some qq c states degenerate with the pion unphysical onset of nuclear matter at µ o 2 m π. Goldstone baryons: bug for QCD, feature for QC 2 D... Calculations with the true complex measure det 2 M nullify effects of qq c states for the vacuum with T =, 2 m π < µ < 3 m N by cancellations among configurations with different signs/phases The Silver Blaze Problem...

6 . p./3 3 Fermion density 2.2 HMC TSMB TSMB(+) Im!. n!!2!3...2 Re!! µ This has been numerically verified, eg. in TSMB simulations of Two Color QCD with N = adjoint staggered quarks. SJH,Montvay,Scorzato,Skullerud, EurPJ C22 (2) 45 The fake transition to a superfluid phase, forbidden by the Pauli Principle, at µ o a.35 disappears once configurations with detm < are included with the correct weight.

7 QC 2 D - the large Nc - limit with gauge group SU(2): detm = detτ 2 Mτ 2 = detm so det is real 2, 2 are equivalent qq baryons and qq mesons lie in same multiplets For µ mπ<<mρ the µ-dependence can be studied using chiral effective theory (χpt) Key prediction: for µ ½mπ a non-zero quark density nq> develops, along with a superfluid diquark condensate qq Textbook BEC formed from weakly-interacting Goldstone qq baryons

8 . p.4/3 Quantitatively, for µ > µ o χpt predicts ψψ ψψ = ( µo µ ) 2 ; n q = 8N f f 2 πµ ( ) µ4 o µ 4 ; qq ψψ = ( µo µ ) 4 [Kogut, Stephanov, Toublan, Verbaarschot & Zhitnitsky, Nucl.Phys.B582(2)477] confirmed by QC 2 D simulations with staggered fermions m=. m=.5 m=. 3. <""> <" tr "> (<""> 2 +<" tr "> 2 ).5 n B µ/m! µ [SJH, I. Montvay, S.E. Morrison, M. Oevers, L. Scorzato J.I. Skullerud, Eur.Phys.J.C7(2)285, ibid C22(2)45]

9 Thermodynamics at T = from χpt quark number density n χp T = 8N f f 2 πµ pressure p χp T = Ω V = µ µ o n q dµ = 4N f f 2 π energy density ε χp T = p + µn q = 4N f f 2 π ( µ4 o ( [KSTVZ] ) µ 2 + µ4 o 2µ 2 µ 2 o µ 4 ) ( µ 2 3 µ4 o µ 2 + 2µ 2 o ) conformal anomaly (T µµ ) χp T = ε 3p = 8N f f 2 π ( ) µ 2 3 µ4 o + 4µ 2 µ 2 o speed of sound v χp T = p ε = NB (T µµ ) χp T < for µ > 3µ o ( µ4 o µ 4 +3 µ4 o µ 4 ) 2. p.5/3

10 This is to be contrasted with another paradigm for cold dense matter, namely a degenerate system of weakly interacting (deconfined) quarks populating a Fermi sphere up to some maximum momentum k F E F = µ n SB = N fn c 3π 2 µ3 ; ε SB = 3p SB = N fn c 4π 2 µ4 ; δ SB = ; v SB = 3 Superfluidity arises from condensation of diquark Cooper pairs from within a layer of thickness centred on the Fermi surface: qq µ 2. p.6/3

11 . p.7/3 2.5 n!pt /n SB.5 "!PT /" SB p!pt /p SB v!pt /v SB µ Q /µ o µ/µ o By equating free energies, we naively predict a first order deconfining transition from BEC to quark matter; eg. for f 2 π = N c /6π 2, µ d 2.3µ o.

12 Simulation Details ( Nf =2 Wilson flavors) Coarse Lattice: 8 3 x6 β=.7 κ=.78 a=.23(5)fm; mπa=.79(); mπ/mρ=.779(4); T=54()MeV O(3) HMC trajectories of mean length.5 on coarse lattice SJH, S. Kim and J.I Skullerud, Eur. Phys. J. C48 (26) 93 O(5) HMC trajectories of mean length.5 on fine lattice Fine Lattice: 2 3 x24 β=.9 κ=.68 a=.86(8)fm; mπa=.68(); mπ/mρ=.8(); T=44(2)MeV SJH, S. Kim and J.I Skullerud, PRD8 (2) 952(R) also have µ-scans on 2 3 x6, 6 3 x2 T=66(3), 88(4)MeV To counter IR fluctuations and maintain HMC ergodocity, we introduce a diquark source term jκ(ψ tr 2 Cγ 5 τ 2 ψ ψ Cγ 5 τ 2 ψtr 2 ) In most results presented here ja=.4 Have just completed µ-scan on fine lattice with ja=.2

13 Computer Effort N cg j=.4 j=.4 V=6!x24 j=.3 j=.2 dt Acceptance aµ The number of congrad iterations required for convergence during HMC guidance rises with µ accumulation of small eigenvalues of M?

14 Equation of State on Fine Lattice (2 3 x24) ! q /! SB.5 n q /n SB p/p SB n q /n SB (a=.23fm) ! (GeV) Identify onset at μ o 36MeV Transition/crossover to quark matter at μq 53MeV nq 4-5fm -3

15 Conformal Anomaly Tµµ = ε-3p (T µµ ) g (T µµ ) q T µµ -. (T µµ ) g = a β a 3β Tr t + s ; LCP N c (T µµ ) q = a κ a κ (4N f N c ψψ ) LCP µ (GeV) Quark and gluon contributions: very similar for μ<μq: NR bound states? differ for μ>μ Q : q, g governed by different statistics? (T μμ ) q changes sharply at μ D 85MeV ε>3p in limit μ

16 Order parameters Superfluid condensate qq.2 j extrapolation scales à la BCS for μq μ μd <qq>/µ 2..4 <qq>/µ 2 Polyakov line Polyakov (a=.23fm) µa µ (GeV) Polyakov line rises from zero at μ μ D Deconfinement at μ D 85MeV n q 6-32 fm -3

17 Into the interior of the T-µ plane a =.8fm a =.23fm <L> !x24 j=.4 6!x24 j=.4 2!x24 j=.3 2!x24 j=.2 2!x6 j=.4 6!x2 j=.4 T (MeV) µ (MeV) aµ Recent data from 2 3 x6, 6 3 x2 (T=66, 88MeV) show that µ D is highly T-sensitive... ΔµD/ΔT - <L> µ=. µ=.575 µ= while β-scans on 2 3 x6 suggest there may be more than one deconfined phase! One superfluid, one normal? ! T 8MeV

18 Quark Number Susceptibility (w/ P. Giudice, J.I. Skullerud) m free =. m free =.67 m free =.333 χ q = T V s 2 ln Z µ 2! nq /! nq free χq χpt /χq SB µ exactly what s expected of a Fermi surface for µq<µ<µd Sensitivity to value of mfree

19 Unexpectedly, χq(µ) does not show same T-dependence as the Polyakov loop 2..8 "=.9, k=.68, j=.4.3 "=.9, k=.68, j= ^3*2 2^3*6 6^3*24.2 6^3*2 2^3*6 6^3*24! nq /µ Polyakov loop aµ aµ Apparently the increase in χq is not associated with deconfinement Qualitatively different from (a) the thermal QCD phase transition (b) analytic/numerical studies on small, cold volumes (the attoworld ) SJH, J. Myers, T.J. Hollowood, JHEP 7 (2) 86, 2 (2) 57

20 Partial Summary QC 2 D has several distinct phases as µ is increased a vacuum phase with nq for µ<µo a superfluid BEC phase described by χpt for µo<µ<µq a superfluid confined quark matter phase for µq<µ<µd deconfined quark matter for µ>µd (perhaps in both superfluid and normal versions) Behaviour for µq<µ<µd resembles the quarkyonic phase postulated by McLerran and Pisarski based on large-n c considerations [L. McLerran and R.D. Pisarski Nucl. Phys. A796 (27) 83] NB: using Wilson fermions we are unable to determine whether the quarkyonic phase is chirally symmetric The deconfining transition at µd is very T-dependent - how many deconfined phases are there?

21 Topological Susceptibility SJH, P. Kenny, PLB7 (2) 373 We have investigated instanton distributions and sizes using cooling.8 N f = 2 N f = 4.6 " #"% "! #"$ !"! #"$ Topological susceptibility shows no structure for Nf = 2 (maybe lattice too coarse?) but appears enhanced in quarkyonic region for Nf = 4 dimensionless plot χ.25 /σ.5 vs. µ/σ.5 Cf. suppression in superfluid phase for Nf = 8 B. Alles, M. D Elia & M.P. Lombardo, NPB752(26)24

22 .5!"#"$%&'!"#"$%('!"#"$%)!"#"$%* N !(a) "# $! For µo<µ<µd the mean instanton size ρi decreases f(!% ~ e -const/!& One-loop Debye screening: Schäfer & Shuryak n I (µ) exp [ RMP 7(998)323 N f ρ 2 Iµ 2] exp [ const ] µ ! In QCD 2+ : Enhancement of U()A breaking first-order region of Columbia plot grows with µ?

23 .5!"#"$%&'!"#"$%('!"#"$%)!"#"$%* N !(a) "# $! For µo<µ<µd the mean instanton size ρi decreases f(!% ~ e -const/!& One-loop Debye screening: Schäfer & Shuryak n I (µ) exp [ RMP 7(998)323 N f ρ 2 Iµ 2] exp [ const ] µ ! In QCD 2+ : Enhancement of U()A breaking first-order region of Columbia plot grows with µ?

24 Quarkonia Results from 2 3 x24 j=.2 Study propagation of heavy QQ (QQ) states through baryonic medium using tree-level, tadpole-improved NRQCD.3.2 M S (µ) M S () M=3. M=4. M=5..3 M3 S (µ) M S (µ) M=3. M=4. M= Mass of singlet s-wave state shows interesting µ-dependence. Also see weak effect in hyperfine splitting Interpret as QQ Qq + qq as nq?

25 p-wave states not well-fitted by a simple pole µ =.75 (m = 3.) µ =. (m = 3.) quarkyonic S P 2.5 deconfined S P Here we plot the propagator ratio CQQ(t;µ)/CQQ(t;) for spin-singlet s- and p- waves states Qualitative difference between quarkyonic and deconfined regimes

26 Summary dense QC 2 D has three distinct transitions/crossovers at μo<μq<μd: Vacuum for μ<μo BEC for μo<μ<μq Quarkyonic phase for μq<μ<μd (are the 2-body bound states of Nc=2 special?) Deconfined phase for μ>μd It s deconfinement, Jim!...but not as we know it? Very temperature sensitive - how many deconfined phases are there? signs of interesting topological structure - instantons enhanced at small µ Use of QQ states as probes of baryonic matter